To prove the conservation of energy is true for the mass-spring system where the mass of spring is not negligible.
Experiment:
Preliminary: Before we start the lab, we have to come up with some
equations of energy for both the hanging mass and the spring. Since we count in
the mass of the spring, the system should have the sum of 5 energies. For
hanging mass, we have KE, GPE. For spring, there are EPE, GPE, and KE.
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| Representation of how GPE spring is calculated |
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| Calculating GPE of spring |
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| Calculating KE of spring |
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| V (end)= velocity measured (by LoggerPro) at the bottom of hanging mass |
1. Mount
a table clamp to the table with a vertical rod then another horizontal rod
attached to the vertical rod. On the horizontal rod, attach force probe and
place it far enough from the table. Place motion sensor on the ground and align
probe and sensor.
2. Calibrate
the force sensor with zero then 100g (9.8N) mass.
3. Measure
the mass of spring. Hang the spring from the force probe. Then, attach 50g mass
to its end and just support it so the spring is non-stretched. Zero the force
sensor
4. Change
the setting to recording data for 3 seconds and test by collecting data by
pulling down the hanging mass.
5. There
are 2 ways to determine spring constant:
- Slowly pull hanging mass downward. Collect this data for 3 seconds at 20times/sec. Obtain Forces vs. stretch, where stretch= initial position-final position. Change same setting for LoggerPro to give Linear fit in a form of F= kx + F0
- Change hanging mass, and record change in position. Let hanging mass pull spring to maximum stretch. Collect data of position (stretch) on LoggerPro. Use this to calculate through: sum of F= kx= mg => k= (mg)/x
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| Setup for this whole lab |
Various Energies involved
1. Hang
250g mass, pull about 10cm downward, let go, then collect data. After 3
seconds, LoggerPro collects, graph velocity vs. time and position vs. time.
2. Make
a new column for data of “stretched” position
3. New
calculated column for all five energy equations from above and another column
for Esum.
Data and Analysis:
Determine the spring constant
We go with the first way by using Linear Fit to find spring constant:
Therefore, k= 8.427 N/m
Various Energies involved
- After pulling and letting go of the system, we get:
- Note that Force, seen here, is directly proportional to position, similarly to Hook's Law: F= kx. At the highest peak of "position" and "Force", the "velocity" is roughly zero. This is where the spring is at maximum stretch. The same case for when force and velocity are at lowest peak, velocity is zero for spring is at maximum compression.
- After pulling and letting go of the system, we get:
Measured data: M-spring= 0.09kg; m-hanging= 0.25kg;
-We determine the non-stretched position of the spring to 0.64cm, a measurement from the top of motion sensor to the bottom of hanging mass. What LoggerPro measure is "position" between these two points. As the mass-spring system oscillate, position changes and we find the stretched/compressed value by calculating: 0.64 - "position".
-New calculated columns for 5 energies. We combine KE and GPE of spring and hanging mass together. So we have 3 energy equations instead of 5, including:
- Putting into LoggerPro, we get:
- The graphs: color-coded: Kinetic energy, Gravitational PE, Elastic PE, and total energy
- The first graph shows relationships between the energies and position. Note that as the system returns to its relaxed position, at 0.64m, Elastic PE decreases and Gravitational PE increases due to increase of height. When comparing all the energies to time in the second graph, we see that GPE is at its lowest point when EPE is at its highest peak. Kinetic energy is at maximum, when GPE and EPE intersect one another (having the same value). In both graphs, total energy stay approximately constant at any point, which is represented by a roughly straight line, since all the other energies compensate one another at different position and time.
Conclusion:
The graph above represents the conservation of energy in the mass-spring system where mass of spring is counted in. Source of uncertainty include the error in measuring the oscillating system by LoggerPro, error in functionality of spring, measurement uncertainty in masses and lengths. Within these listed possible error, we can still conclude that there's conservation of energy in this mass-spring system.
- The first graph shows relationships between the energies and position. Note that as the system returns to its relaxed position, at 0.64m, Elastic PE decreases and Gravitational PE increases due to increase of height. When comparing all the energies to time in the second graph, we see that GPE is at its lowest point when EPE is at its highest peak. Kinetic energy is at maximum, when GPE and EPE intersect one another (having the same value). In both graphs, total energy stay approximately constant at any point, which is represented by a roughly straight line, since all the other energies compensate one another at different position and time.
Conclusion:
The graph above represents the conservation of energy in the mass-spring system where mass of spring is counted in. Source of uncertainty include the error in measuring the oscillating system by LoggerPro, error in functionality of spring, measurement uncertainty in masses and lengths. Within these listed possible error, we can still conclude that there's conservation of energy in this mass-spring system.
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